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Extragalactic Astronomy
Didier Fraix-Burnet, Marc Thuillard, Asis Kumar Chattopadhyay
To cite this version:
Didier Fraix-Burnet, Marc Thuillard, Asis Kumar Chattopadhyay. Multivariate Approaches to Clas- sification in Extragalactic Astronomy. Frontiers in Astronomy and Space Sciences, Frontiers Media, 2015, 2 (3), pp.00. �10.3389/fspas.2015.00003�. �hal-01184937�
Extragalactic Astronomy
Didier Fraix-Burnet1,2,∗, Marc Thuillard3, Asis Kumar Chattopadhyay4
1Univ. Grenoble Alpes, IPAG, F-38000 Grenoble, France
2CNRS, IPAG, F-38000 Grenoble, France
3La Colline, 2072 St-Blaise, Switzerland
4Department of Statistics, University of Calcutta, India Correspondence*:
Didier Fraix-Burnet
Institut de Plan ´etologie et d’Astrophysique de Grenoble (UMR 5274), BP 53, F-38041 Grenoble C ´edex 9, France, [email protected]
2015, Frontiers in Astronomy and Space Sciences 2, 3 ABSTRACT
Clustering objects into synthetic groups is a natural activity of any science. Astrophysics is not an exception and is now facing a deluge of data. For galaxies, the one-century old Hubble classification and the Hubble tuning fork are still largely in use, together with numerous mono- or bivariate classifications most often made by eye. However, a classification must be driven by the data, and sophisticated multivariate statistical tools are used more and more often. In this paper we review these different approaches in order to situate them in the general context of unsupervised and supervised learning. We insist on the astrophysical outcomes of these studies to show that multivariate analyses provide an obvious path toward a renewal of our classification of galaxies and are invaluable tools to investigate the physics and evolution of galaxies.
Keywords: Clustering – Classification – Galaxies – Multivariate Analysis – Phylogenetic Methods
1 INTRODUCTION
Astrophysics has always adopted specific strategies to classify a relatively modest amount of diversity and has much counted on the physics to define the discriminant parameters. This discipline is now facing the need for sophisticated statistical tools to tackle the astronomical number of observed and catalogued objects and the increasing number of observed properties that describe them.
The debate about the usefulness of the morphological classification of galaxies is a rather old one and is still alive. Sandage (Sandage, 2005), a proponent of a (morphological) classification driven by the data, noticed that the Hubble classification and the Hubble tuning fork have not yet been replaced by anything else despite the efforts of the proponents of a classification driven by the physics (e.g. Conselice, 2006).
It also has been recognised to have many flaws: it is a qualitative, subjective and visual approach, difficult to use for distant galaxies, it is based solely on the visible morphological parameter while galaxies are complex and evolving systems and while we have at our disposal morphologies from X-rays to radio wavelengths, spectra, chemical compositions, stellar populations, central black hole masses, kinematics of stars and gas...
However this debate may not address the right question since from a classification point of view, a clas- sification must be driven by the data, and thus be multivariate (e.g. Fayyad et al., 1996). Consequently, adapted tools must be used which are not well known to astronomers in general. Nevertheless, numer- ous studies have been published during the last thirty years or so, especially since the beginning of the XXIst century. In this paper we would like to present these different approaches in the general context of unsupervised (clustering) and supervised (classification) learning.
Clustering approaches gather objects according to their similarities either through the choice of a dis- tance metric or using some adequate criteria for deciding to which cluster some object belongs. There is a huge class of techniques that partition the data into a pre-defined number of clusters. A well-known algorithm is the k-means (MacQueen, 1967; Ghosh and Liu, 2010).
Another family of clustering techniques uses a hierarchical representation of the pairwise distances between objects in terms of a number of parameters (variables), through a bottom-up algorithm that constructs a tree by relating the closest objects together before relating these first clustering to closest clusters or objects, and so forth until the whole sample is exhausted. The final number of groups is then chosen by cutting the tree at a fixed distance level. The branches of the tree, called a dendrogram, may or may not represent relationships between the objects.
Originally, phylogenetic methods are designed to build a graph representing the evolutionary relation- ships between species (see reviews in Felsenstein, 2003; Makarenkov et al., 2006). Each node of the graph indicates a transmission with modification mechanism that creates two or more species inheriting from a common ancestor. More generally, a phylogenetic approach can be viewed as an unsupervised clustering approach in which relationships are provided. As a consequence, phylogenetic techniques are particularly versatile and powerful methods for building classification trees. They can be understood in the framework of the graph theory (Semple and Steel, 2003).
There are two kinds of phylogenetic methods, based either on the pairwise distances (or dissimilarities) computed from the parameters describing the objects, or on these parameters themselves.
The distance-based methods build the tree entirely from the distances, putting forward the global sim- ilarities between the objects. The friends-of-friends algorithm is relatively famous in astrophysics (e.g.
More et al., 2011, and references therein). Also known as the single linkage or Nearest Neighbor algo- rithm, it is mathematically related to the Minimum Spanning Tree technique which looks for the simplest graph connecting the objects under study (Gower and Ross, 1969; Feigelson and Babu, 2012). A more sophisticated approach used in phylogenetic studies is the Neighbor-Joining Tree technique (Saitou and Nei, 1987; Gascuel and Steel, 2006).
In the parameter-based methods, the parameters are called characters which in astrocladistics corre- spond to the parameters associated to the physical measurements of some properties of the objects. The parameter-based methods evaluate all possible trees that can be constructed with the objects, and select the tree(s) corresponding to an optimization criterion. The process is thus based on the distribution of the parameter values.
Parameter-based methods can describe a larger variety of evolutionary scenarios and are thus more gen- eral that the distance-based methods. But this is at the cost of a larger computation time which quickly becomes prohibitive. Mathematically, formal connections between parameter- and distance-based meth- ods are developed in the case of continuous parameters (e.g. Thuillard and Fraix-Burnet, 2009, 2015), explaining why both kinds of methods are successfully used in phylogenetic studies.
Among the parameter-based techniques, cladistics is the most famous one. Invented in the 1950’s by William Hennig (Hennig, 1965), its principle looks simple: two (or more) objects are related if they share a common history, that is they possess properties inherited from a common ancestor. In practice, a cladis- tic analysis asks for the objects under study to be described by evolutionary characters (parameters or descriptors) for which at least two states are defined: one is said to be ancestral, the other one is said to be derived. The derived state corresponds to an innovation in the evolution and is assumed to have been acquired by an unidentified ancestor. This is the transmission phase of inheritance making descendants
look similar to their parents. The accidents in this process are called modifications and generate diversity.
This transmission with modification process was invoked by Darwin to explain the observed hierarchi- cal organisation of the biological diversity. Several approaches have been developed to search for the best tree representation using Maximum Likelihood, some Bayesian approaches or Maximum Parsimony.
In Maximum Parsimony, one searches for the tree representation of the data with the smallest number of evolutionary steps to explain the data. But in essence, any entity, be it biological or not, evolving with a transmission with modification process can be a priori studied by Maximum Parsimony, provided evolutionary states can be defined for the characters.
A more general representation of relationships are given by networks even though their interpretation is quite complex, but they can be approximated by several trees.
In this review, we do not intend to present all possible techniques in both supervised and unsupervised learning. Rather, we focus on the astrophysical published studies made with the objective of discovering structures in a data set, in other words a new clustering and possibly a new classification of galaxies, beyond the traditional Hubble morphological scheme. We refer the reader to the complete review by Ball and Brunner (2010) on data mining tools used in astrophysics for further information and references in particular on the separation of sources or the classification of galaxies into morphological types. Our paper is mainly devoted to unsupervised classification (clustering) and presents the phylogenetic methods which are not included in Ball and Brunner (2010). In addition we insist on the astrophysical outcome and the new insights that such studies have brought to our knowledge on galaxy physics and diversity.
Part of this paper is inspired from De et al. (2013) which compares the applicability of some of the clustering techniques on the basis of Gaussian and non Gaussian astronomical data sets. Here we do not make such a comparison.
The paper is organized as follows. The first section presents a frequent prerequisite to data mining, the dimension reduction (Sect. 2). This approach has been heavily used in the extragalactic literature to identify groups in the reduced component space, the motivation being mainly for automatic classification in large data sets. The second section describes the important difference between this motivation, called (supervised) classification, and the clustering (unsupervised classification) which is the main topic of this paper (Sect. 3). We also discuss shortly the concept of similarity between objects.
Partitioning methods divide the sample into distinct groups. This can be made with hard or soft bounds depending on whether the membership is a probability or not (see e.g. Andrae et al., 2010). Thek-Nearest Neighbor (Sect. 4), Support Vector Machine (Sect. 5) and k-means (Sect. 6) methods are of the first kind.
The fuzzy clustering approach (Sect. 7) belongs to the soft partitioning techniques and often extends the applicability of the previous methods. The Information Bottleneck approach is able to provide both kinds of classification (Sect. 8).
These partitioning methods require the number of classes as an input. Some other techniques try to fit some distributions to the data set, the optimization process providing the number of groups best fitting the data. These techniques are based on mixture model (Sect. 9) and wavelet (Sect. 10) methods.
A different category of clustering approaches establishes relationships between the objects and derive the groups from the generated graph. The first such category are the hierarchical methods (Sect. 11) which build a tree based on the pairwise distances. Different cuts on the tree result in different numbers of classes. These cuts can be chosen on the basis of objective arguments but also may vary according to the goal of the analysis since the tree provides a synthetic view of the structures within the data set, instead of just the group memberships. Another kind of graphs are the networks produce by the Minimum Spanning Tree method (Sect. 12). The last kind of relationships are evolutionary relationships. This is the domain of the phylogenetic techniques, a very wide subject of bioinformatics. We here present only the Maximum Parsimony (cladistics), Neighbor-Joining Tree Estimation and Outer Planar Networks that have been applied in the context of galaxies (Sect. 13).
2 DIMENSION REDUCTION APPROACHES 2.1 METHODS
When the data set is large (both in terms of number of variables and number of observations) one may first apply some appropriate dimension reduction technique and then perform clustering on the reduced data set.
One must keep in mind that the discriminant usefulness of distances is lost in high dimension parameter spaces since distances tend to become similar (one of the aspects of the “curse of dimensionality”).
Principal component analysis (PCA)
In this technique, given a data set of observations on correlated variables, an orthogonal transformation is performed to convert it into a set of uncorrelated variables called the principal components. The number of principal components is less than or equal to the number of original variables. This transformation is de- fined in such a way that the first principal component has the largest possible variance. One rule of thumb is to consider those components whose eigen values are greater than one in the reduced space. Principal components are guaranteed to be independent only if the variables are jointly normally distributed.
Independent component analysis (ICA)
Principal component analysis, Factor Analysis, Projection Pursuit are some popular methods based on linear transformation. But ICA is different because it looks for the components in the representation that are both statistically independent and non Gaussian. ICA separates statistically independent components, which are the original source data, from an observed set of data mixtures. All information in the multivari- ate data sets are not equally important. There is often a need for extraction of the most useful information.
ICA extracts and reveals useful hidden factors from the whole data sets. ICA defines a generative model for the observed multivariate data, which is typically given as a large database of samples. Contrarily to PCA, the components are not imposed to be orthogonal.
Independent Component Analysis (Comon, 1994), model assumes the form
X =AS (1)
whereXis a data matrix,Ais the non-singular mixing matrix,S is matrix of independent components.
A−|is the unmixing matrix. The main goal of ICA is to estimate the unmixing matrixA−|. It is assumed that the data variables are linear or non-linear mixtures of some latent variables and the mixing system of equation 1 can be written as
Xi =ai1S1+ai2S2+...+ainSn, i= 1,2, ..., n (2) The Si are mutually independent and aij are the entries of the non-singular matrix A. Here n is the number of parameters (variables). For performing ICA, the data set has to be whitened in the sense that correlations in the data have to be removed.
There is no rigorous method to determine the optimum number of ICs. For instance, the number of independent components can be taken to be equal to the number of principal components with eigen values greater than 1 (Albazzaz and Wang, 2004). As most of the data sets in Astrophysics are likely to be non Gaussian, ICA can be successfully used in many situations (Chattopadhyay et al., 2013a,b).
2.2 APPLICATIONS
PCA technique was applied in a few papers in the 1970s and 1980s with the goal of finding the main parameters explaining the variance among galaxy samples. For instance, Watanabe et al. (1985) used four parameters (diameter, magnitude, mean surface brightness and mean concentration index) and found
that two principal components explains 97% of the total variance in their sample of all morphological types, in agreement with other studies. While Watanabe et al. (1985) do not find differences in the two- dimension PC plane between elliptical and disk galaxies, Whitmore (1984) more explicitly looks for an objective classification of galaxies: “ The fact that there are so many different classification systems for galaxies...demonstrates that we are still searching for the fundamental properties.”. Using more parameters (up to 15) they agreed with the other studies on two components explaining most of the variance, and tentatively identify them as scale and form. They do not devise a new classification scheme, but rather identify different correlations depending on the position of the galaxies on the 2D diagram.
Chattopadhyay and Chattopadhyay (2006) also found two components in a PCA analysis of samples of spiral galaxies with extended rotation curves. They constructed new “fundamental planes” with these components, pinpointing the most important physical factors. They also performed a multiple stepwise regression analysis of the variation of the overall shape of the rotation curves and find that it is mainly determined by the central surface brightness, while the shape purely in the outer part of the galaxy (beyond the optical radius) is mainly determined by the size of the galactic disk. Such a regression is interesting to predict still unobserved values for some parameters, and is improved by the reduction of the dispersion in the principal component space.
Peth et al. (2015) used PCA as a simple way to reduce the dimensionality, break internal degeneracies and find the natural distributions of data in the parameter space characterizing the structures and shapes of galaxies that they study. These principal components are then used to classify the shape of galaxies through a hierarchical clustering technique (see Sect. 11).
Several studies (e.g. Connolly et al., 1995) used PCA both as a dimension reduction and as a tool for classification of spectra of galaxies. Spectra are characterized by a high number of attributes (the wavelengths) that are not independent since a spectrum is made of a continuum spectrum from stars plus absorption and emission lines from the gas. PCA has in principle the power to identify the minimum number of spectra to combine in order to obtain the observed diversity. (Connolly et al., 1995) used a variant of the PCA technique, the Karhunen-Love transform, which allows for weighting differently some parts of the spectra. They not only find that two eigenspectra are necessary to account for most of the variance of the spectra of galaxies, but the distribution of classes in the two-parameter space is one-dimensional. They proposed a scheme of ten classes, some corresponding to the broad morphological types Sa, Sb, S0 and E, while the six others are starburst objects. Their work was intended to be used by spectral surveys to classify automatically the observations.
In a similar scope of general classification of galaxies, one must mention the attempt by Scarlata et al.
(2007) to build a morphological automated classification of galaxies, the ZEST catalog, using PCA (see Coppa et al., 2011) but the parameters used are criticized by Andrae et al. (2010). This illustrates the importance of the selection of the parameters for a multivariate clustering or classification analysis which at some point may appear arbitrary and subjective. A special care should be brought to this initial step through the analysis of the data set itself with dedicated data mining tools.
Another instance is the classification established by Conselice (2006) using a PCA analysis together with a Spearman Rank correlation test to better understand the parameters of the data set. His approach is to use the PCA on some set of parameters and then understand the physics of the principal components.
So the PCA shed light on the underlying physics from which a classification scheme can be built. He finds three dimensions for this scheme, with the mass (scale), the star formation (spectral type) and the interactions/mergers (degree of dynamical disturbances). This should remind that PCA is not a clustering technique per se, it provides a new representation of the data from which a clustering may be performed.
Indeed the work by Conselice (2006) proposes new relationships between the morphological classes. His scheme appears as a more physical replacement of the 2D Hubble tuning fork diagram.
The Principal Component Analysis assumes a linear combination of the parameters, a rather strong assumption. Taghizadeh-Popp et al. (2012) have used a non-linear PCA, the Principal Curve analysis,
“which can be seen as a nonparametric extension of linear PCA. The principal curve is the curve follow- ing the location of the local mean in the multi-dimensional cloud of data points.” They obtain a drastic
dimension reduction with a one-dimension parameter space (the Principal Curve) which they divide arbi- trarily into 20 groups of equal densities. They compute a distance (the arc length) that ranks the galaxies so providing a natural and objective way of ordering, partitioning and classifying the rich zoo of galaxies in the nearby universe. Taghizadeh-Popp et al. (2012) do not include luminosity nor mass in the process in order not to bias the study of the luminosity function as a function of the arc length. This is debatable but they are right in saying that it would induce a bias since these parameters will define a strong axis of variance in the PCA. Nevertheless would it be possible to classify galaxies without their mass? Could massive galaxies have the same history as less massive ones? This shows that the choice of the parameters is never so obvious, and generally related to the choice of the technique used as well. The interesting point is that they recover known trends in the physics of galaxies, but more importantly they can identify new kinds of galaxies pointing out particular physical processes and histories of galaxies. These discoveries can only be made by multivariate analyses.
Folkes et al. (1996) applied PCA on spectra of low signal-to-noise ratio mainly as a dimensionality reduction technique. The few principal components are then used to train a neural network in order to classify galaxies into the five broad morphological types. Even though this approach is efficient for big data sets, it appears limited to normal galaxies since they find that a new classification scheme must be used where unusual features are present in the spectra.
The ICA analysis is still less common than PCA for the study of galaxies. At least two studies have been published, an ensemble learning for ICA (Lu et al., 2006) and a mean field independent component analysis (Allen et al., 2013). In the first case, 1326 synthetic spectra have been used coming from Single Stellar Population models. They select 74 ”sufficiently” different spectra from these (using an objective criterion) since the ensemble learning part converges very slowly. The ICA analysis yields six most signif- icant components, and the 1326 spectra are fitted on these components. Each component represent a basic element behind the spectra of galaxies, and they find that each of them can be associated closely to one or a few stellar types plus some peculiar line properties. These six components are then used on real galaxy spectra to derive the stellar contents like starlight reddening, stellar velocity dispersion, stellar content, and star formation history. Even though PCA is much faster, it does not provide this important information because of the orthogonality constraint that does not allow the components to be non-negative.
Allen et al. (2013) used the mean field ICA which is a probabilistic ICA using a prior to constrain the components. They find that ten components (divided into five continuum and five emission components) are required to produce accurate reconstructions of essentially all narrow emission-line galaxies to a very high degree of accuracy. Using these ten components on a large sample of Sloan Digital Sky Survey (SDSS) galaxies, they identify the regions of parameter space that correspond to pure star formation and pure active galactic nucleus (AGN) emission-line spectra, and produce high S/N reconstructions of these spectra.
In a similar fashion, Hurley et al. (2014) applied the Non-negative Matrix Factorization technique which has been developed for blind source separation problems. Unlike PCA, this technique imposes the condi- tion that weights and spectral components are non-negative that is also possible in the ensemble learning approach for ICA described above (Lu et al., 2006). This more closely resembles the physical process of emission in the mid-infrared region studied in this work, resulting in physically intuitive components.
They find seven such components, including two for active galactic nucleus emission, one for star forma- tion, and one for the rising continuums at longer wavelengths. They show that the seven components can be used to separate out different types of objects (see Sect. 9) and to separate out the emission from AGN and star formation regions and define a new star formation/AGN diagnostic which is consistent with all mid-infrared diagnostics already in use but has the advantage that it can be applied to MIR spectra with low signal-to-noise ratio or with limited spectral range.
3 SUPERVISED AND UNSUPERVISED LEARNING 3.1 DISTANCES/DISSIMILARITIES
A lot of learning techniques require a dissimilarity measure. Among them, the distances obey the well- known triangular properties and define a metric. In hierarchical clustering, the distances mainly come from a very general distance known as the Minkowski’s distance or thepth norm, which may be defined as follows. For two points P= (x1, x2, ..., xn)and Q= (y1, y2, ..., yn)in thendimensional space, the pth norm is given by
Lp =
n
X
i=1
|xi−yi|p
!1/p
(3) Forp= 1, it gives the Manhattan distance (L1norm). Forp= 2, it reduces to the Euclidean distance (L2
norm). Also forp =∞, the Lp norm results in Chebyshev distance. In hierarchical clustering, Euclidean and Manhattan distances are mainly used. But these measures are applicable only to continuous data. For categorical or binary data other distances must be used but will not be addressed in this paper.
It may be noted that the selection of the appropriate distance matrix for clustering problems completely depends on the physical situation.
3.2 SUPERVISED LEARNING (CLASSIFICATION)
Supervised learning technique may be viewed as a mapping between a set of input variables and an output variable. This mapping is applied to predict the outputs for unseen data. The main characteristic of super- vised learning is the availability of annotated training data. It supervises the learning system to instruct on the labels to associate with training examples. These labels are known as class labels in classification problems. Supervised learning induces models for the training data and these models are then used to classify other unlabeled data. Two most popular supervised learning techniques are the Nearest Neighbor (Sect. 4) and the Support Vector Machines (Sect. 5) classifiers.
3.3 UNSUPERVISED LEARNING (CLUSTERING)
The unsupervised learning or clustering seeks some pattern in the data set by starting from the raw data with or without any distributional assumption regarding the underlying distribution. The three main cat- egories of this kind are (i) connectivity based clustering (like hierarchical clustering, see Sect. 11), (ii) centroid based clustering (like k-means, see Sect. 6) and (iii) density based clustering (like DBSCAN or more generally kernel density estimation).
An overview of these approaches can be found in D’Abrusco et al. (2012) with many references of appli- cations in astrophysics. Most of the methods that we present in the following are unsupervised clustering.
The reason is that the multivariate analyses of galaxies essentially are either supervised approaches based mainly on dimension reduction techniques (mostly PCA, see Sect. 2) or unsupervised methods to discover new classification schemes of galaxies which are really objective and multivariate.
4 NEAREST NEIGHBOR 4.1 METHOD
Thek-Nearest Neighbor (NN) algorithm is very intuitive. It starts from a training set for which we have the class labels. In order to make a prediction about a new observation, one looks at the labels of its k
Figure 1. In this k-Nearest Neighbor illustration with k = 5, the central black square more probably belongs to the blue class.
nearest neighbors and uses a majority vote to make the prediction (Fig. 1). As the number of neighbors used in making the prediction increases, the decision boundaries become smoother, the bias increases, but the variance decreases.
4.2 APPLICATIONS
Ball et al. (2007) explored the k-Nearest Neighbor technique for determining photometric redshifts in petascale databases using 55 746 quasar spectra from the SDSS. The algorithm is trained on a repre- sentative sample of the data. The main result is that the comparison between the photometric and the spectroscopic redshifts shows no region of catastrophic failure where the two derived values differ a lot, contrarily to other methods used to derive photometric redshifts.
5 SUPPORT VECTOR MACHINE 5.1 METHOD
Support Vector Machine (SVM) aims to find the hyperplane that best separates two classes of data through an optimization method. Instead of using just a standard orthogonal basis, SVM uses many functions to describe good separating surfaces. The input data are viewed as sets of vectors, and the data points closest to the classification boundary, determined from a training sample, are the support vectors. SVM funda- mentally separates two classes of objects which is probably a limitation in its use for the classification of galaxies.
They use optimization methods to find surfaces that best separate categories. Their key innovation is to express the separating surfaces in terms of a vastly expanded set of basis functions. Instead of using just a standard orthogonal basis, SVMs use many basis elements.
5.2 APPLICATIONS
SVM has been used by Huertas-Company et al. (2008) for the morphological classification of galaxies from the COSMOS survey. The training sample is a limited sample classified visually using a 12- dimensional volume, including 5 morphological parameters, and other characteristics of galaxies such as luminosity and redshift. The objective is to be able to classify automatically the results of big sur- veys. However, the result seems a little bit disappointing since it can only separate between the two broad
classes of early- and late-type galaxies, with an error of about 20%, even though this is better than other methods generally used.
6 K-MEANS 6.1 METHOD
The k-means algorithm (MacQueen, 1967; Ghosh and Liu, 2010) is a partitioning approach that starts withkcentroids,kcorresponding to the number of clusters given a priori. It then assigns each data point to the closest centroid and when the clusters are built, the newk centroids are computed and the process iterates until convergence (Fig. 2). The result depends very much on the initial centroids. Repeating the analysis with several initial choices is always a good idea, but consistency is not guaranteed if the data do not contain distinguishable and roughly spherical clusters. Some strategies have been devised to guess the best initial choice for the centroids (e.g. Sugar and James, 2003; Tajunisha and Saravanan, 2010) and many indices are available in the packageN bClust(Charrad et al., 2014) of R (Team, 2014).
A variant called the k-medoids algorithm (Kaufman and Rousseeuw, 1987; Reynolds et al., 2006) chooses data points as centers (medoids) and is known to be more robust to noise and outliers.
Figure 2. A typical result of a k-means analysis in which the clusters (four here) are clearly distinguishable (from Fraix-Burnet et al., 2010).
6.2 APPLICATIONS
The k-means algorithm has been used in the context of stars (e.g. Gratton et al., 2011; Simpson et al., 2012), galaxies (e.g. Fraix-Burnet et al., 2010; S´anchez Almeida et al., 2010; Fraix-Burnet et al., 2012) or Gamma-Ray Bursts (Chattopadhyay et al., 2007).
S´anchez Almeida et al. (2010) performed a k-means analysis of a large number (788 677) of spectra from the SDSS. Each spectra is a collection of about 4 000 wavelengths, making the full data set very computationally demanding for a direct k-means. They thus decided to limit the spectra to a priori infor- mative regions, reducing the number of “parameters” to 1637. Their analysis is affected by the dependence of the result on the seed. They say that estimation tools for the number of clusters could not be applied because of the sample size. Using some criteria, they end up choosing randomly one classification having
28 classes. The result looks more like a continuum distribution of spectra, and even if not shown, over- lapping between classes is important. This questions the validity of the k-means approach in this case as another k-means analysis of the same sample has shown (De et al., 2014).
Multivariate k-means analyses of smaller sample of galaxies with the aim of discovering new classes of galaxies have been performed as a complement to other clustering methods by Fraix-Burnet et al.
(2010) with the four parameters of the fundamental plane, and by Fraix-Burnet et al. (2012) with six parameters selected from 23 available. In the latter case, the selection of the parameters is made through different statistical tools, in order to find a parameter subspace in which a robust clustering of the data is present. This leads to the important result that several very different clustering techniques yield compatible clusterings, giving good confidence to the result. The astrophysical implications are numerous since a new classification is established and the average properties and the correlations varies from group to group and often differ from those of the global sample. However, the interpretation benefited from the relationships between the classes established by the phylogenetic method used in these works and discussed in Sect. 13.
Even though the clusters are similar, the absence of these links in the k-means results is clearly missing.
Chattopadhyay and Karmakar (2013) performed a k-means analysis of a large sample of dynamically hot stellar systems from globular clusters to giant ellipticals, in quest of the formation theory of ultra compact dwarf galaxies (UCDs), using three parameters (logarithm of stellar mass, logarithm of effective radius and stellar mass to light ratio). The number of clusters, five, is given by the optimum criterion of Sugar and James (2003). The classification of UCDs provides some new clues to the long discussed hypothesis that these objects may be formed either as massive globular clusters or have an origin similar to nuclei of dwarf galaxies.
7 FUZZY CLUSTERING 7.1 METHODS
In non-fuzzy or hard clustering, data is divided into crisp clusters, where each data point belongs to exactly one cluster. In fuzzy clustering, the data points can belong to more than one cluster, and associated with each of the points are membership grades which indicate the degree to which the data points belong to the different clusters. Many algorithms exist, many of them being extension of hard clustering algorithms.
One example is the fuzzy C-means which is very similar to the k-means (Sect. 6) but adding a weight between 0 and 1 to each point characterizing its probability to belong to a given group, and a degree of fuzziness of the groups.
7.2 APPLICATIONS
Coppa et al. (2011) studied the bimodality of galaxies which comes from a double peak distribution in some scatter plots, particularly in color-color diagrams. The origin of this bimodality and the relationship between the two broad classes, “red” and “blue” or “late type” and “early type”, is still not understood.
Evolution is probably involved, but then what is the status of the overlapping regions called “the green valley”? To know whether this bimodal distribution is an intrinsic property of galaxies and their evolu- tion, multivariate analysis must be used since it appears in several scatter plots. Coppa et al. (2011) use an unsupervised fuzzy partition clustering algorithm applied on the principal components of a PCA anal- ysis. They use eight parameters, two coming from spectra, one from photometry and five describing the morphology. They keep three principal components to perform the clustering analysis which proceeds in two steps: a modified fuzzy k-means algorithm to guess the memberships and the cluster centroids, and a second algorithm (fuzzy modification of maximum likelihood estimation) to achieve optimal fuzzy partition (see references in Coppa et al., 2011).
They decide to identify three clusters, blue, red and green, somewhat giving up the fuzzy nature of their study. In addition, they name the clusters after previous classifications, even though “the ’early type cluster
is not intended to be made up of pure passive galaxies; rather, it is composed also by bulge-dominated weakly-star forming objects.” This is a quite confusing practice especially because they discover some new kinds of objects which are invaluable for our understanding of the physics and evolution of galaxies.
Bayesian approaches can also be seen as soft classification as illustrated for instance in the separa- tion between star forming galaxies and Active Galactic Nuclei (AGNs) in Norman et al. (2004) to avoid confusion between different kinds of objects.
8 INFORMATION BOTTLENECK TECHNIQUE 8.1 METHOD
The Information Bottleneck Method (Tishby et al., 2000) is a simple optimization principle for a model- free extraction of the relevant part of one random variable with respect to another. The algorithm is extremely general and may be applied to different problems in analogous ways. A great advantage of this unsupervised clustering technique is that it avoids the arbitrary choice of the distance and provides a natural quality measure for the resulting classification.
Using the mathematical notations of Slonim et al. (2001) that applied this technique to galaxies, the optimal classification is given by maximizing the functional:
L[p(c|g)] =I(C; Λ)−β−1I(C;G) (4) whereCrepresents the classes,Gthe galaxy sample andΛthe spectral wavelengths.I(C; Λ)andI(C;G) are the mutual information betweenC;Λ andC;G. β−1 is the Lagrange multiplier attached to the com- plexity constraint. For β → 0the classification is non-informative, and for β → ∞ the representation becomes arbitrarily detailed.
8.2 APPLICATIONS
Slonim et al. (2001) explain that by normalizing the total photon counts in each spectrum to unity, we can consider it as a conditional probability, the probability of observing a photon at a specific wavelength from a given galaxy. The ensemble of spectra can thus be seen as a conditional probability distribution function that allows to undertake the information theory-based analysis. For any desired number of classes, galaxies are classified such that the information content about the spectra is maximally preserved.
The number of classes is an issue in most unsupervised clustering techniques, and the information bottleneck shares this difficulty too. Slonim et al. (2001) note that ’the true or correct number of classes may be an ill-defined quantity for real data sets and the number should be determined by the desired resolution, or preserved information“. However one should be careful to use objective arguments based only on statistics, since the physical interpretation should come at the end to tell whether or not the result is interesting.
The main results of this study is the demonstration that an objective and automated technique can yield a classification of spectra which is very physical, in the sense that it recovers results obtained more clas- sically, but is able to discover other classes and correlations between physical parameters. An interesting point in their study is that they applied the same techniques to two samples, one observed and one sim- ulated. The good agreement between the two clusterings shows that the models of galaxy evolution are sensible. This is a good approach to test the models by statistically comparing two populations using multivariate data sets.
9 MIXTURE MODELS 9.1 METHODS
Most partitioning methods use a distance to define the clusters. In model-based clustering methods, each cluster can be represented by a parametric distribution, the data set being thus considered as a mixture of such model distributions (Qiu and Tamhane, 2007). The parameters include the mixing proportions or the prior probabilities of the clusters since the true cluster memberships of the observations are unobserved.
The optimization relies on the likelihood of the weighted linear combination of the cluster distributions through the Expectation-Minimization (EM) algorithm. Clustering is done by applying the maximum posterior (Bayes) rule. The process yields a soft classification (probability of membership) and a fit to each cluster distribution.
The mixture model approach also provides expected misclassification probabilities. It requires the num- ber of clusters to be known, which can be for instance estimated with the tools developed for the k-means analysis (Chattopadhyay et al., 2009, Sect. 6).
9.2 APPLICATIONS
Davoodi et al. (2006) find four Gaussian distributions best fit the color distribution of 16 698 extragalactic infrared sources. They use this result to propose a classification scheme (Ca toCd) of galaxies that reveal a greater variety of galaxy types than usual spectral energy distribution fitting techniques that strongly depends on the quality of the template model components. Interestingly, Davoodi et al. (2006) use their soft classification to identify outliers (rare galaxies or transient phases) by summing up the four probability density functions for each object.
Hurley et al. (2014) used the seven components they have found with a dimension reduction approach (Sect. 2) to define a parameter space in which they apply an unsupervised Gaussian Mixture Model clus- tering algorithm in order to provide a classification tool. This clustering approach is a fuzzy approach since clusters describe a probability density function indicating how likely a galaxy could be found in any one of the clusters. Eight clusters are found which are consistent with previous classifications. Strangely enough, these clusters are named according to the classical classification through a majority rule. We may ask why use an unsupervised technique if one believes in an existing ”true“ hard classification?
10 WAVELET ANALYSIS 10.1 METHOD
The wavelet transform is a well known signal analysis technique widely used in many research areas. Its key property is the ability to provide a multi-resolution approximation of a given input signal through a prototype functionΨ:
W(s, r) = Z
f(t) 1
√sΨ
t−r s
dt (5)
wherescharacterizes the scale andrthe translation factor. The prototype function, also called the mother wavelet, is continuous in both time and frequency and serves as the analysing window.
With this definition, wavelets appear as a parametric-model decomposition of a data set using some basis functions. They could then be used for dimension reduction and/or classification (Thuillard, 2001).
Shapelets are a scaled version of two-dimensional Gauss-Hermite polynomials and form a set of com- plete basis functions that are orthonormal on the interval[−∞,∞]. Shapelets are thus suited to decompose images. For galaxies, their use is limited to high signal-to-noise data and rather regular galaxies since they
are gaussian-shaped and spherical (Andrae et al., 2010). The composition is an automatic and objective representation of galaxy morphologies.
Other multiresolution methods have been proposed, like for instance the hierarchical Markov models extended for the multispectral astronomical image segmentation (Collet and Murtagh, 2004).
10.2 APPLICATIONS
Wavelets can be used to decompose galaxy spectra into several features that can then be used to classify the spectra. In this sense they serve as a dimension reduction technique but contrarily to PCA or ICA the basic elements (features) can be chosen to be physically meaningful, representing the three components of spectra: the continuum, the emission and absorption lines (e.g. Starck et al., 1997; Liu et al., 2005).
Andrae et al. (2010) review how an automatic classification of galaxy morphologies could be done us- ing shapelets. Their goal is not to devise a new classification, since it is extremely difficult to parametrise arbitrary galaxy morphologies apart from the question that the morphology is only one property of galax- ies. To address the parametrisation problem, they use shapelets and then define the distance as the angle spanned by their (normalised) coefficient vectors of the shapelets:
d(xi, xj) = arccos(xi·xj) (6)
They then use a soft (fuzzy) clustering algorithm with the similarity matrix given by:
Wmn ∝ (d(xi, xj)/dmax)α
s (7)
withdmax being the maximum distance between any two objects in the given data sample, andα >0and s > 1being free parameters that tune the similarity measure. This probabilistic clustering technique uses the graph theory in which the similarity elementsWmnare the weights of the edges.
They also evaluate the impact of hard clustering methods on the estimation of the parameters charac- terising the classes depending on the level of overlapping. This is an important point to keep in mind in all hard (non-fuzzy) approaches to clustering, be it by hand or algorithmic. They even suggest that the processes of galaxy evolution and observations tend to invalidate hard clustering approaches.
They do not go into the details of the astrophysical interpretation, but they clearly demonstrate the advantages of such sophisticated approaches for automatic morphological classification of a huge number of galaxies. However, as they rightly say, “a lot of work is still needed on the interpretation of the results.”
11 HIERARCHICAL CLASSIFICATION METHODS 11.1 METHODS
The hierarchical classification method builds a hierarchy of clusters. Two main approaches to form the hierarchy are agglomerative and divisive. In the agglomerative approach each observation is considered as a cluster and pairs of clusters are merged as one moves up the hierarchy (see Fig. 3). The most similar objects are grouped first and those initial groups are merged ultimately into single cluster according to some proximity measure. These proximity measures are based on either similarities or dissimilarities (distances). In the divisive analysis approach all observations at first are grouped in one cluster, and splits are performed recursively as one moves down the hierarchy. Here an initial single group of objects is divided into two subgroups such that the objects in one subgroup are far from the objects in the other.
These subgroups are further divided into dissimilar groups until there are as many subgroups as objects.
Figure 3.An example of a dendrogram. Distance between two horizontal branches is going from left to right. The two dashed lines illustrates two cuts yielding 9 or 3 clusters.
In order to decide which clusters should be combined or where a cluster should be split, a distance matrix is required. The distances used for hierarchical clustering are mainly Euclidean and Manhattan for continuous type data. In order to find distances between clusters different linkages like single linkage, complete linkage, average linkage etc are used. Note that the nature of the final clusters totally depends upon the choices of distances and linkages.
It is interesting to note that if the metric used is the single linkage, then this method is similar to the Minimum Spanning Tree technique (Sect. 12).
11.2 APPLICATIONS
Peth et al. (2015) applied a Ward hierarchical agglomerative clustering to classify galaxies in distinct groups using the first three principal component eigenvectors. In this kind of approach the number of groups is chosen after the analysis. Peth et al. (2015) selected ten groups as a compromise between too many small groups which might appear as too specific, and too large ones that would smear out the true diversity of the objects. They also try to define boundaries to these groups in the PC-space by fitting a convex hull around the points within each groups in order to classify future new observed objects. How- ever, a Nearest-Neighbor or SVM technique could be used in this purpose without the need to compute a convex hull which is a rigid boundary. It is important to recall that a classification is never definitive and would probably evolve with the inclusion of new objects, as it has been for instance the case for the S0 (lenticular) morphological class of galaxies which were not present in the original Hubble classification.
One of the main results of the studies by Peth et al. (2015) is a refined and objective classification of structures and morphologies of the galaxies in their samples. The ten groups are analyzed separately to de- rive their properties and their probable evolutionary status and history. Their scheme separates quenched compact galaxies from larger, smooth proto-elliptical systems, and star-forming disc-dominated clumpy galaxies from star-forming bulge-dominated asymmetric galaxies. It also reveals a higher fraction of bulge-dominated galaxies than visual classification or one based on the Sersic index.
Decision trees are a practical use of hierarchical clustering. S´anchez Almeida et al. (2012) propose a decision tree to classify galaxy spectra according to some general features that usually serves as a classification of galaxy properties. They use the decision tree on their previously ASK classes determined with the k-means technique (Sect. 6, S´anchez Almeida et al., 2010). Somehow, in this way, they classify their new classes on another classification.
Suchkov et al. (2005) have applied an oblique decision tree classifier on the homogeneous multicolor imaging data base of the SDSS, the classifier being trained on subsets of objects (stars and galaxies) whose nature is precisely known via spectroscopy. Each node in the decision tree is a criterion on one parameter, defining an hyperplane parallel to one of the axis. In an oblique decision tree, the criterion is based on a (linear) combination of parameters, so the tree is no more parallel to any of the axes in the parameter space. In Suchkov et al. (2005) the classifier is composed of ten oblique decision trees and the final decision is made by votes which yield a class probability distribution for a given object. The main result of their study is to show the ability of this approach to automatically classify objects from the photometry instead of the spectroscopy which is harder to obtain and analyse, and accurately predict the redshifts of both normal and active galaxies. This can increase considerably the samples required to analyse statistically the evolution and diversity of galaxies, their properties and their correlations.
12 MINIMUM SPANNING TREE 12.1 METHODS
The Minimum Spanning Tree (MST) is mathematically related to the single linkage clustering, known to astronomers as the friends-of-friends algorithm or Nearest Neighbor algorithm (Gower and Ross, 1969;
Feigelson and Babu, 2012). A spanning tree is an acyclic, connected graph Gwhich is a set (V, E)of vertices (nodes) and edges (branches) together with a functionw :E → Rthat assigns a weightw(e)to each edgeeinE. The minimum spanning tree (Fig. 4), is the spanning treeT minimizing the function :
w(T) = X
e∈T
w(e) (8)
If the weightsw(e)are distinct, then the solution is unique. A number of algorithms have been developed to solve exactly the Minimum Spanning Tree problem. The first algorithm is attributed to Boruvka (1926).
Other popular algorithms are Prim’s, Krukal’s and the Reverse-Delete algorithms that all find solutions in polynomial time. The above algorithms also work at higher dimensions in which case the Euclidean L2 or the Manhattan L1 distances are generally used.
Minimum spanning trees have found applications in phylogeny, computer vision, and cytology just to name some domains. It has been used in astrophysics, and maybe very early since a large number of constellations defined by early civilizations are also shown to correlate well with a Minimum Spanning Tree (Dry et al., 2009).
Figure 4.An illustration of a Minimum Spanning Tree linking ten nodes.
12.2 APPLICATIONS
The Minimum Spanning Tree technique has been heavily used to determine the galaxy clusters in order to map the spatial distribution of the baryonic matter, a visible signature of the gravitational structure of the Universe shaped by the Dark Matter (Barrow et al., 1985; Bhavsar and Splinter, 1996). This is not an application to clustering in the sense of classification, but this is a spatial clustering. Indeed the MST approach has been strongly adapted to the particular constraint of cosmological observations: the exact position along the line of sight is only approximately given by the redshift. We do not discuss any further this application which is not in the main scope of this paper.
We know of only one use of MST for galaxy classification. Ascasibar and Sanchez-Almeida (2011) applied this technique to understand their ASK classification of SDSS spectra obtained with the k-means method (Sect. 6, S´anchez Almeida et al., 2010). They find that the majority of the spectral classes are distributed along a well-defined branch going from the earliest to the latest types, with optically bright active galaxies forming an independent branch that intersects the main sequence exactly at the transition between early and late types. This description is already an interpretation of the 23 ASK classes that present a regular distribution of their spectra as already mentioned in Sect. 6, so that the very linear structure of the MST tree is not surprising. However, the approach is interesting because this is a rather simple and objective method to obtain relationships between classes.
13 PHYLOGENETIC METHODS
Basically, all galaxies share a common origin which is the gathering of baryonic matter as a self gravitating object. This baryonic matter was very primitive and has subsequently being enriched and diversified by several generations of stars and many transforming processes like galaxy interactions and mergers. There are thus obvious evolutionary relationships between different kinds of galaxies as immediately understood by Hubble when he discovered galaxies and established his famous tuning fork diagram. Taking into account the galaxy diversity of morphologies known at that time, he built a phylogenetic tree in which the relationships are due to the evolution of the stellar orbits which, he thought, should flatten with time because of the dynamical friction. Even though we now know that this process cannot be accomplished
in a time shorter than the known age of our Universe, this tuning fork diagram is still used to represent galaxy diversity.
Somewhat strangely enough, phylogenetic analyses of galaxy diversity has not been attempted again for a century. This is probably because the data did not allow much progress into this direction. But we now have huge multivariate databases and it seems timely to reconsider this question. We here present only a few techniques, those that have been already used on astrophysical data sets.
13.1 METHODS
Before describing some of the most important methods, let us point out that the development of phyloge- netic methods has been hindered till the 2000s by very heated discussions on the philosophical merits of the different approaches. It is only in recent years that most of the barriers between the different schools of thoughts could be overcome by a new generation of researchers. Recently a new picture of phyloge- netic methods is emerging. It becomes nowadays increasingly clear that all the different approaches can be discussed within a common framework including distance- and character-based approaches, and that this theoretical framework applies both to phylogenetic trees and networks.
There are two main categories of methods: the distance-based and the character-based. The “characters”
are traits, descriptors, observables, parameters or properties, which can be assigned at least two states characterizing the evolutionary stage of the objects for that character. For continuous parameters, these states can be obtained through discretization.
Distance-based Approaches: Neighbor Joining Tree Estimation
For distance-based approaches, Neighbor-Joining is the most popular approach to construct a phyloge- netic tree. The Neighbor Joining Tree Estimation (NJ, Saitou and Nei, 1987; Gascuel and Steel, 2006) is based on a distance (or dissimilarity) matrix. This method is a bottom-up hierarchical clustering methods.
It starts from a star tree (unresolved tree). A “corrected” distanceQ(i, j)between objectsiandjfrom the data set ofnobjects, is computed from the distancesd(i, j):
Q(i, j) = (n−2)d(i, j)−
n
X
k=1
d(i, k)−
n
X
k=1
d(j, k) (9)
The branches of the two objects with the lowestQ(i, j)are linked together by a new nodeuon the tree.
This node replaces the pair(i, j)in the subsequent iterations through the distance to any other objectk:
d(u, k) = 1
2[d(i, k)−d(i, u)] + 1
2[d(j, k)−d(j, u)] (10)
Neighbor-Joining minimizes a tree length, according to a criteria that can be viewed as a Balanced Minimum Evolution (Gascuel and Steel, 2006). For a tree metrics, Neighbor-Joining furnishes a simple algorithm to reconstruct a tree from the distance matrix. There is a large literature on how to best ap- proximate a metrics by a tree metrics (see for instance Fakcharoenphol et al., 2003). Neighbor-Joining is justified if the difference between the original distance matrix and the distance matrix describing the X-tree obtained with Neighbor-Joining is not too large.
Character-based Approaches: Cladistics, Maximum Parsimony, Maximum Likelihood...
Cladistics has been associated in the 80’s to the search of a maximum parsimony tree. Maximum Parsi- mony is a powerful approach to find tree-like arrangements of objects (Fig. 5). The drawback is that the analysis must consider all possible trees before selecting the most parsimonious one. The computation complexity depends on the number of objects and character states, so that too large samples (say more than a few thousands) cannot be analyzed. The Maximum Parsimony algorithm can take uncertainties or
Figure 5.A example tree obtained with cladistics, represented here as unrooted. When a root is chosen, the tree takes the shape of hierarchical trees.
unknowns into account by evaluating the different possibilities allowed by the range of values and select- ing among them the one that provides the smallest score. In the case of unknown parameters, the most parsimonious diversification scenario provides a prediction for the unknown values.
In recent years the definition of cladistics has been extended to the classification of taxa (individuals or species) defined by characters on a rooted tree. In biological applications, a phylogenetic tree describes the possible evolution of a taxon corresponding to the root. The root may either be a real taxon or be inferred from the descendant taxa. The success of a cladistics analysis much depends on the behavior of the parameters. In particular, it is sensitive to redundancies, incompatibilities, too much variability (reversals), and parallel and convergent evolutions. It is thus a very good tool for investigating whether a given set of parameters can lead to a robust and pertinent diversification scenario.
If a set of characters exactly defines a phylogeny, then the phylogeny is called perfect. In practical applications, the available characters seldom define a perfect phylogeny. A supplementary measure of the deviation to a perfect phylogeny is necessary to determine how well a candidate tree fits the characters.
In the standard approach to parsimony, the score sp of a tree corresponds, after labeling of the internal nodes, to the minimum number of edges(u, v)withc(u)6=c(v),c(u)being the character state at nodeu.
The tree with the minimum score is searched for with some heuristics (Felsenstein, 1984). The maximum parsimony approach can be directly extended to continuous characters or values. To each internal node is associated a real valuef(u). The score s of a tree equals the sum over all edges of the absolute difference between those values:
s= X
e=(u,v)E
|f(u)−f(v)| (11)
Robinson (1973) has shown that for a tree defined by continuous characters, a maximum parsimony score is reached for values of the internal nodes belonging to the set of values (or states) defined on the leaves.
The main method to search for the best tree representation of data beyond Maximum Parsimony include Maximum Likelihood. We note this technique which has never been applied to astrophysics in the context of classification but may be a pertinent approach. The problem here is that an evolutionary model must be used, and naturally the result will depend significantly on it. Maximum Likelihood is used standardly in biology, and it may be possible that astrophysicists could also have well constrained physical models of the evolution of galaxies and their properties. The phylogenetic tree of Maximum Likelihood is the tree for which the observed data are most probable (Williams and Moret, 2003). Distance-based approaches are also often quite appropriate for reconstructing a phylogenetic tree from continuous characters. Distance- based approaches are fast and can be used for data exploration and for the selection of the most appropriate variables.
Cladistics when applied to domain outside of biology, like in astrocladistics, refers more generally to the classification of objects by a rooted or an unrooted tree (Fig. 5). In that case, the tree represents possible relationships between taxa. The search of the best tree described by a set of characters on a set of objects (or taxa in phylogeny) can be done by several different approaches. The most popular methods are the one using Maximum Parsimony or Maximum Likelihood. For continuous parameters, the software program TNT (Goloboff et al., 2008) is also quite popular to reconstruct trees from characters. As an alternative, the data can be discretized through appropriate binning.
As mentioned earlier, a new picture of phylogenies is emerging after the understanding that phylogenies on multistate characters reduce through a conceptually simple grouping of the characters into a phylogeny on binary characters. For binary characters, both distance- and character-based approaches are equivalent.
This approach opens new perspectives as it furnishes also a bridge between character-based phylogenies and split networks or more precisely outer planar networks.
Outer planar networks
Outer planar networks permit the simultaneous representation of alternative trees with reticulations, and are thus generalizations of trees (Huson and Bryant, 2006). In order to understand the connection between outer planar networks and phylogenetic trees, one has to explain succinctly what is called a split on a circular order of the taxa. A circular order on a phylogenetic tree corresponds to an indexing of the n end nodes according to a circular (clockwise or anti-clockwise) scanning of the end nodes. A split on a circular order of the taxa is a partition of the objects into two disjoint sets (Fig. 6).
Figure 6.A circular order for objects A to G, with their pairs of binary states, arranged according to the circular consecutive-ones condition. The two lines show two different split, one between (0,*) and (1,*), the other one between (*,0) and (*,1).
For multistate characters, a split can be defined after transformation of each multistate character into a binary character. For each pair of states (A,B), a subset of states containing the state A is attributed the 1 state and the complementary subset including the subset B is given the binary state 0. If the transformation can be done on each states and characters (for details see Thuillard and Fraix-Burnet, 2015) so that each binary character fulfills the circular consecutive-ones condition, then the data can be described exactly by an outer planar network. By definition the circular consecutive-ones condition are fulfilled if for any binary state, the taxa with the 1 state are consecutive on the circular order (Fig. 6).
Splits in an outer planar network (Fig. 7) furnish neighboring relationships between objects. Objects sharing a common property, as defined by splits, are consecutive in a circular order. Outer planar net- works can be regarded as a generalization of phylogenetic trees. An outer planar network reduces to a phylogenetic tree if for each pair of binary characters, the so-called 4-gamete rule is fulfilled. The 4- gamete rule states that for each pair of binary characters there is at least one of the 4 possible gametes (either (1.0), (0,1), (1,1) or (0,0)) that is missing.
Figure 7.An example of an outer planar network showing the eight splits of the eight parameters s1 ...
s8.
For distance-based approach, the circular consecutive-ones conditions have to be replaced by the fulfillment of the Kalmanson inequalities. For taxa indexed according to a circular order, the dis- tances between a reference noden and the path i− j are gathered in the distance matrix
n Yi,jn
o with Yi,jn = 12(di,n+dj,n−di,j), di,n being the pairwise distance between leave iand noden. This distance matrix fulfils the so-called Kalmanson inequalities:
Yi,jn ≥Yi,kn , Yk,jn ≥Yk,in (i≤j ≤k) (12)
Bandelt and Dress (1992) have shown that if a distance matrix n Yi,jno
fulfils Kalmanson inequalities, then the distance matrix can be exactly represented by a split network or by an X-tree. The program SplitTrees4 (Huson and Bryant, 2006) permits to construct outer planar networks from a distance matrix.
In practice, the perfect order is not known or not feasible. The difference between the perfect order and the order one obtains with a given data set is called the contradiction. The minimum contradiction analysis (Thuillard, 2007, 2008) finds the best order one can get. It is a powerful tool for ascertaining whether the parameters can lead to a tree-like arrangement of the objects (Thuillard and Fraix-Burnet, 2009). Using the parameters that fulfil this property, the method then performs an optimisation of the order and provides groupings with an assessment of their robustness.
We believe that outer planar networks will gain importance in applications outside of biology as they furnish a real alternative to the standard classification methods.
13.2 APPLICATIONS
Farrah et al. (2009) have used a bayesian framework to compare and group 102 ultra-luminous infrared galaxy spectra and yield a network diagram which is used to define three groups. An evolutionary descrip- tion of these galaxies is proposed from the properties of these groups. Even though their method is not a phylogenetic technique per se since the relationships are constructed after the clustering analysis, this work illustrates the potential need of phylogenetic tools in astrophysics.
The use of phylogenetic approaches in astrophysics has been pioneered and pursued through the denom- ination of astrocladistics (Fraix-Burnet et al., 2006b,c,a). Applications have been successfully performed for galaxies (Fraix-Burnet et al., 2010, 2012), globular clusters (Fraix-Burnet et al., 2009; Fraix-Burnet and Davoust, 2015) and Gamma-Ray bursts (Cardone and Fraix-Burnet, 2013).
The phylogenetic approaches used on galaxy samples are clearly oriented towards a multivariate and evolutionary classification of galaxies (Fraix-Burnet et al., 2010, 2012). To this end, several statistical analyses (PCA, k-means, cladistics and minimum contradiction analysis) are used to select the set of parameters that yields a robust classification according to several clustering analyses (k-means, cladistics and Minimum Contradiction Analysis). Six parameters were so selected among the 23 available: the central velocity dispersion, the disc-to-bulge ratio, the effective surface brightness, the metallicity, and the line indices NaD and OIII. The agreement of the clustering obtained by different techniques reinforces largely the result. The cladistics tree (cladogram) is used for the interpretation since it also provides the relationships between the groups.
These relationships are hypothesized as being evolutionary so that the placement of the groups on some diagrams reflects the evolution of the properties and their correlations. For instance, the famous funda- mental plane is not universal at all, this 3D correlation clearly depends on the diversification level of the group: the correlation becomes tighter when the history of a galaxy is more complex. Other well-known correlations, like Mgbvs the velocity dispersion, indeed disappear within the groups but is created by the alignment of the groups in the scatter plot. This strongly suggests that these correlations (known as scal- ing relations) are statistical and caused by a hidden confounding factor, which is possibly the evolution (Fraix-Burnet, 2011).
The new classification is rather easily interpreted with all the parameters available and by comparison with numerical simulations. The galaxies within a given group share a common history, that is a sequence of transforming events (collapse, interaction, harassment, merger...) that Fraix-Burnet et al. (2012) are able to identify.
Outer planar or split networks have also been applied on galaxy samples (Thuillard and Fraix-Burnet, 2009) even though it is for a theoretical illustration of an optimisation approach to fulfil as much as possi- ble the Kalmanson inequalities (Eq. 12). Nevertheless, a classification is obtained on this limited sample of 100 galaxies and with only three parameters. The main splitting character is the surface brightness (Brie)
